Paper: arXiv 2610.40404

Authors: Danila Litvinov

Abstract

We establish multidimensional exponential convergence of the lifted Bass gradient flow in martingale optimal transport. Let $μ\le_{cx}ν$ be an irreducible pair, with compact convex target support, source support strictly inside the target support, and target density bounded above and bounded away from zero almost everywhere on its interior. For every bounded $σ(X)$-measurable initial state, the flow converges exponentially in $L^2$ to the global minimizer with the prescribed mean, and the functional value gap decays exponentially. Both bounds hold at every nonnegative time with the exact initial errors as prefactors. The quantitative step is strong convexity on sets with a common $L^\infty$ bound and a fixed mean. We prove it by interpolating entropic endpoint potentials, using two conditional copies to cancel the unknown potential increment, and obtaining a Hessian lower bound uniform in the regularization parameter. Removing the regularization yields curvature for the original functional; the known qualitative theory supplies a common bound for the orbit. This extends the one-dimensional exponential-convergence result of Backhoff-Veraguas, Pammer, and Schachermayer. The main result is formalized in Lean 4. AI was used in preparing the paper (see Section 8).

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly theoretical and mathematically dense analysis of exponential convergence for a specific gradient flow. While the mathematical rigor and novelty are strong, there is no empirical validation or backtesting, placing it firmly in the ‘Lab Rats’ quadrant. The formal verification in Lean 4 significantly boosts reproducibility.

Research Flowchart

  flowchart TD
    A[Research Goal: Establish Multidimensional Exponential Convergence of Lifted Bass Gradient Flow] --> B{Key Methodology: Strong Convexity & Interpolating Entropic Endpoint Potentials};
    B --> C[Inputs: Irreducible Pair μ ≤_cx ν, Compact Convex Target Support, Source Support < Target Support, Target Density Bounded];
    C --> D[Computational Processes: Lean 4 Formalization, AI for Paper Preparation];
    B --> E{Key Method Detail: Using Two Conditional Copies to Cancel Unknown Potential Increment};
    E --> F[Quantitative Step: Hessian Lower Bound Uniform in Regularization Parameter];
    F --> G[Outcomes: Exponential L2 Convergence to Global Minimizer, Exponential Functional Value Gap Decay, Bounds with Exact Initial Errors];
    A --> G;